Revisiting the Kalman’s Conjecture to Stabilize the Motion of a DC Motor in the Presence of Stribeck Friction via PID Control
In absence of Stribeck friction, the application of PID to control the motion of a DC motor may lead to stable oscillation. The stable oscillation, in particular cases called limit cycle oscillation, occurs in low-speed control as well as in position control. In this research, the mitigation of stable oscillation has been performed from the perspective of Kalman’s conjecture. From the point of view of Kalman’s conjecture, the minimum required proportional gain in order to stabilize low-speed control has been derived. In addition, the minimum derivative gain and the maximum integrator gain necessary to stabilize position control are proposed. This research is complementary to previous research, which has applied the circle criterion for generating a stable PID strategy. The novelty of this research is that the PID strategy proposed in this research provides a more relaxed integrator gain requirement than the previous one, especially for low proportional gain. The findings of this research are beneficial for PID blind tuning since all stability requirements have been derived analytically.
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Tsuruta, K. Sato, K. Fujimoto, T. Ushimi, and Nobuhiro, High precision positioning control for table drive system using pid controller with nonlinear friction compensator, Proceeding of the 4th International Conference on Leading Edge Manufacturing in 21st Century, Fukuoka; Japan, November 2007.
O. Rubes, M. Brablc, and Z. Hadas, Nonlinear vibration energy harvester: Design and oscillating stability analyses, Mechanical Systems and Signal Processing, Vol. 125: 170-184, 2019.
J. Llibre and X. Zhang, X. (2019). The non-existence, existence and uniqueness of limit cycles for quadratic polynomial differential systems. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 149 (Issue 1): 1-14, February 2019.
Hayat and P. Shang R. M. Saeed, A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope, Automatica, Vol. 100: 52-60, February 2019.
S. Stramigioli and M. van Dijk, Energy Conservative Limit Cycle Oscillation, IFAC Proceedings Volumes, Vol. 41 (Issue 2): 15666-15671, 2008.
X. Sui and Q. Ding, Bifurcation and stability analyses for a pad-on-disc frictional system, International Journal of Non-Linear Mechanics, Vol. 107: 112-125, December 2018.
R. H. A. Hensen, M. J. G. van de Molengraft, Friction induced hunting limit cycles: An event Mapping Approach, Proceeding of 2002 American Control Conference (IEEE Cat. No. CH37301), Vol. 3, pp. 2267-2272, Anchorage, AK, USA, May 8-10 2002.
L. Marton, On Analysis of limit cycles in positioning systems near Stribeck velocity, Mechatronics Vol. 18 (Issue 1): 46-52, February 2008.
M. R. Popovic, G. Liu, and A. A. Glodenberg, Experimental study on low velocity friction compensation and tracking control, Journal of Automatic Control, Vol. 13(Issue 2): 17-22, January 2003.
Tumbuan, T., Nurprasetio, I., Indrawanto, I., Abidin, Z., Stable PID Control Strategy to Remove Limit Cycle Due to Stribeck Friction on DC Servo Motor, (2018) International Review of Automatic Control (IREACO), 11 (4), pp. 208-216.
R. D. Robinett III and D.G. Wilson, What is a limit cycle?, International Journal of Control, 81 (Issue 12):1886-1900, October 2008.
K, Ciesielski, The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century, Open Mathematics, Vol. 10 (Issue 6): 2110-2128, June 2012.
M. Borre and H. Flashner, Periodic Solutions for Flexible Structures Under Relay Feedback Control With Time Delay, ASME 2012 5th Annual Dynamic Systems and Control Conference joint with the JSME 2012 11th Motion and Vibration Conference, Vol. 2, pp. 607-616, Fort Lauderdale, Florida, USA, October 17–19, 2012.
H. Gao, Y. Song, and M. Li, The Analysis for the Reason of Limit Cycle Generated by Friction in Servo System, 2010 International Conference on Computer, Mechatronics, Control and Electronic Engineering, pp. 88-91, Changchun, China, August 2010.
R. Guo, X. Jiang, and H. Wang, Boundary Value Problems, Vol. 2019, (Issue1):5, January 2019.
C. Canudas‐de‐Wit, J. Aracil, F. Gordillo, amd F. Salas, The oscillations killer: a mechanism to eliminate undesired limit cycles in a class of nonlinear systems, International Journal of Robust and Nonlinear Control, Vol. 24 (Issue1): 39-53, January 2014.
J. C. A. de Bruin, A. Doris, N. van de Wouw, W.P.M.H. Heemeles, and H. Nijmeijer, Control of mechanical motion systems with non-collocation of actuation and friction: a Popov criterion approach for input to state stability and set valued of nonlinerities. Automatica 45 (Issue 2): 405-415, February 2009.
T. Naderi, D. Materassi, G. Innocenti, R. Genesio, and M.V. Salapaka, Revisiting Kalman and Aizerman conjectures, 2017 Proc. IEEE 56th Annual Conference on Decision and Control pp 523-528, Melbourne, VIC, December 2017.
G. A. Leonov, V. O. Bragin, and N. V. Kuznetsov, Algorithm for constructing counter examples to the Kalman problem, Doklady Mathematics 82 (Issue 1): 540-542, August 2010.
Taki El-Deen, A., Mahmoud, A., R. El-Sawi, A., Optimal PID Tuning for DC Motor Speed Controller Based on Genetic Algorithm, (2015) International Review of Automatic Control (IREACO), 8 (1), pp. 80-85.
S. M. Rozali, M.F. Rahmat, and A.R. Husain, Performance Comparison of Particle Swarm Optimization and Gravitational Search Algorithm to the Designed of Controller for Nonlinear System, Journal of Applied Mathematics, Vol. 2014, Article ID 679435, 9 pages, 2014.
A. Gossul and H. Giacomini, Some applications of the extended Bendixson-Dulac theorem, In: Ibáñez S., Pérez del Río J., Pumariño A., Rodríguez J. (eds.) Progress and Challenges in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, Vol 54. Springer, Berlin, Heidelberg, 2013.
S. Jeon, Integrator leakage for limit cycle suppression in servo mechanisms with stiction, ASME Journal of Dynamic Systems, Measurement and Control, Vol. 134 (Issue 3): 34502-8 pages, April 2012.
S. Srang and M. Yamakita, Estimation of discontinuous friction using continuous-discrete unscented Kalman filter for adaptive compensation, 2013 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, pp. 429-435, Wollongong NSW, July 2013.
George, R., Hasanien, H., Al-Durra, A., Badr, M., Model Predictive Controller for Performance Enhancement of Automatic Voltage Regulator System, (2018) International Journal on Energy Conversion (IRECON), 6 (6), pp. 208-217.
Magdy, G., Shabib, G., Abdel Elbaset, A., Kerdphol, T., Qudaih, Y., Bevrani, H., Mitani, Y., A Novel Design of Decentralized LFC to Enhance Frequency Stability of Egypt Power System Including Wind Farms, (2018) International Journal on Energy Conversion (IRECON), 6 (1), pp. 17-29.
Eid, A., Abdel-Fadil, R., Abdel-Salam, M., Performance and Power Quality Improvements of MEA Power Distribution Systems using Model Predictive Control, (2017) International Review of Aerospace Engineering (IREASE), 10 (1), pp. 31-41.
Bouallegue, S., Khoud, K., Integral Backstepping Control Prototyping for a Quad Tilt Wing Unmanned Aerial Vehicle, (2016) International Review of Aerospace Engineering (IREASE), 9 (5), pp. 152-161.
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